How Do Cube Roots Work?

Date: 11/04/2003 at 13:01:17
From: Gina 
Subject: i don't understand cubed square roots?

I do not understand how to do a square root problem when it is asking 
you to do something with a cubed root.  For example:      
    ______
\3 / 216  = ? 
 \/

I don't understand what I am supposed to do with the 3.

Date: 11/04/2003 at 17:24:43
From: Doctor Douglas
Subject: Re: i don't understand cubed square roots?

Hi Gina.

Thanks for writing to the Math Forum.

You are correct, you are being asked to find the cube root of 216. 
This is similar to the case of a square root, in which you have to
find the number X such that X * X is equal to the given number.  I
assume that you're familiar with this type of problem (e.g. sqrt(25) =
?, Answer:  5 or -5).

For the cube root (of 216), you are looking for a number such that 

   X * X * X = 216

This is what the notation
       _____
  /\3 / 216  = X
    \/

means.  To find X, you can use a guess-and-check method with a
calculator.  If you are interested in the method for computing a cube
root by hand without a calculator, you should check out the following
web page from our archives:

  Cube Root by Hand
    http://mathforum.org/library/drmath/view/52605.html 

I hope this helps explain what is going on!

- Doctor Douglas, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 11/04/2003 at 17:33:20
From: Doctor Riz
Subject: Re: i don't understand cubed square roots?

Hi Gina -

Thanks for writing.  When you take a square root of something, you are 
trying to find the number that when you SQUARE it you get the number 
inside the radical sign.  For example, the square root of 9 is 3 
because when you square 3 you get 9 ( 3^2 = 3*3 = 9 ).  Similarly, the 
square root of 25 is 5 because 5^2 = 5*5 = 25.

When you take a cube root, you are trying to find the number that when 
you CUBE it you get the number inside the radical sign.  For example, 
the cube root of 8 is 2 because 2^3 = 2*2*2 = 8.  The cube root of
1000 is 10 because 10^3 = 10*10*10 = 1000.  The cube root of 216 is 6 
because 6^3 = 6*6*6 = 216.

The little 3 you see tucked into the radical sign indicates that it's
a cube root.  We could write a little 2 in the same place for a square 
root, but we agree not to bother, so if you see a radical sign without 
a little number you know it's a square root.  

If you saw one with a little 4 in there, that would mean a 4th root.  
That tells you to look for the number that when you raise it to the
4th power you get what's inside the radical.  For example, the 4th
root of 81 is 3 because 3^4 = 3*3*3*3 = 81.

The key thing to realize is that the radical sign itself just tells
you that you are dealing with a root of some sort.  It's the little
index number that tells you what KIND of root you are finding.  No
index number means it's a square root, 3 means cube root, 4 means
fourth root and so on.

Most calculators have a square root key, and many of today's
calculators also have a key where you can specify what kind of root
you want to take.  

One final comment--you may know that you can't take a square root of a
negative number, at least not when using real numbers.  For example, 
there is no real square root of -9 because (3)^2 = 3*3 = 9 and (-3)^2
= (-3)*(-3) = 9.  Whether you square 3 or -3 you get +9 both times.  
There is no number you can square and get -9.

But with cube and other odd roots, you can take them of negative
numbers.  We saw that the cube root of 8 was 2 since 2^3 = 2*2*2 = 8.  
What if it's the cube root of -8?  Note that (-2)^3 = (-2)*(-2)*(-2) = 
-8.  When you multiply an odd number of negatives, the answer is 
negative.  That's why you can take odd roots of negative numbers.

Hope this helps--write back if you are still confused!

- Doctor Riz, The Math Forum
  http://mathforum.org/dr.math/